ON KLEIN'S COMBINATION THEOREM

ON KLEIN'S COMBINATION THEOREM BY

BERNARD MASKIT(i)

The combination theorem of Klein, Der Prozess der Ineinanderschiebung, in [4], was first stated essentially as follows. Let Gx and G2 be finitely generated discontinuous groups of Möbius transformations, and let Dy and D2 be fundamental domains for Gy and G2, respectively. Assume that the interior of Dy contains the boundary and exterior of D2, and that the interior of D2 contains the boundary and exterior of Dy. Then G, the group generated by Gt and G2, is again discontinuous, and £) = D1nD2isa fundamental domain for G. Proofs of the above theorem appear in most of the standard references (for

example, [2, pp. 56-59], [3, pp. 190-194]). These proofs use different definitions of "fundamental domain", none of which are very general. In this paper, we give a new version of this theorem, which uses fundamental sets rather than fundamental domains, and which does not have the restriction that Gy and G2 are finitely generated. We also have somewhat different goemetric hypotheses. The new conditions involve the existence of a certain Jordan curve. In Chapter HI there is an example which shows that such a condition is needed. This example corresponds, in the formulation given above, to the case that the boundaries of Dy and D2 are not disjoint. One of the conclusions of the combination theorem is that G is the free product of Gy and G2. In [5] I proved a weak generalization of this theorem for the case that G is the free product of Gy and G2 with an amalgamated subgroup. In this paper, there is a stronger generalization, in which the amalgamated subgroup is

cyclic. In order to state the main theorem of this paper, certain definitions and notations are needed. These are given in Chapter I, which also contains some basic facts about Kleinian groups. Chapter II contains the formulation of the main theorem and the proof. Chapter III contains the example mentioned above. It was proven in [5] that a rather wide class of Kleinian groups can be constructed from very simple groups using the construction that appears in this paper, and another construction. This other construction will be discussed in a subsequent

paper. Received by the editors March 3, 1965. 0) Part of this research appeared in the author's doctoral dissertation at New York University. The research was supported by the National Science Foundation through grants

NSF-GP779, NSF-GP780, NSF-GP2439, and a graduate fellowship.

499

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

500

BERNARD MASKIT

[December

The two constructions mentioned above are also used by Nielsen-Fenchel [1] to construct all finitely generated Fuchsian groups, starting with certain "elementary" groups. This problem was originally suggested to me by L. Bers. I would like to take this opportunity to thank him, and L. Keen, for many informative conversations.

I. Basic facts. 1. We denote the extended complex plane, or Riemann sphere, by E. A Möbius transformation is a mapping z -» (az + b)(cz + d) ~\ where a, fo,c, d are complex

numbers with ad —be # 0. A group G of Möbius transformations

is said to be discontinuous at the point z,

if there is a neighborhood U of z, such that g(U) O U = 0, for all geG, other than the identity; i.e. if x, ye U, geG, g(x) = y, then x = y and g = l. A Kleinian group is a group of Möbius transformations which is discontinuous at some point z. Throughout this chapter, the letter G will denote a Kleinian group. The following notation will be used throughout this paper. If S cz 2 and H is some set of Möbius transformations, then

n(S,//)= hell \JKS).) 2. Given a Kleinian group G, there are two natural subsets of S which one can associate with G. The regular set R(G) consists of those points of £ at which G is discontinuous. The limit set L(G) consists of those points z for which there is a point z0, and a sequence {g„}, of distinct elements of G, with gn(z0) -» z. One easily sees that these sets are invariant under G, that R(G) is open, that

L(G) is closed, and that R(G) n L(G) = 0. 3. A set D is called a partial fundamental

set (PFS) for G, if

(a) D*0, (b) D cz R(G), and

(c) g(D) nD = 0, for all geG,g^

1.

If, in addition to (a), (b), and (c), D satisfies property

(d)Y[(D,G)= R(G), then D is called a fundamental set (FS) for G. Conditions (c) and (d) are the usual conditions for a fundamental set. Condition (c) asserts that no two points of D axe equivalent under G, and condition (d) asserts that every point of R(G) is equivalent to some point of D. We remark that, using the definition of discontinuity, condition (c) can be restated as follows : if x, ye D, geG, g(x) = y, then x = y and g = I. 5. In this section we prove three basic lemmas about Kleinian groups. Lemma 1. Let x be some point ofE, and let {gn} be a sequence of distinct elements of G Then A = {z|g„(z)-»x} is both open and closed in R(G).


1965]

ON KLEIN'S COMBINATION THEOREM

501

Proof. If z e An R(G), then we can find a neighborhood U of z where g(U) n U = 0 for every geG other than the identity, and U is the interior of a circular disc. Then, for every n, g„(U) is again the interior of a circular disc. Since the g„ are all distinct, g„(U) n gm(U) = 0 for all n^m, and so, in terms of the metric on the sphere, the area of gn(U) converges to zero. It follows that the diameter of gn(U) converges to zero, and so g„(y)->x, for every ye U; i.e. A is open in R(G). Now let Zjt-> z e R(G), where each zk e A n R(G). Pick a neighborhood U of z, as above. For k sufficiently large, zke U, and so, by the above argument,

z e A ; i.e. A is closed in R(G). Lemma 2. Let z„-*zeR(G), and let {g„} be a sequence of distinct elements ofG, with g„(z„) -►x. Then gn(z) -►x. Proof. As in the proof of Lemma 1, let U be a neighborhood of z where U is the interior of a circular disc, and g(U) nU = 0 for all geG other than the identity. Then, for n sufficiently large, z„e U. Since the diameter of g„(U) converges to zero, gn(z) -* x.

The following lemma was essentially proven by Ford [2, pp. 39-41] : Lemma 3. Let z0eR(G) and let {g„} be a sequence of distinct elements of G. Assume that g„(z0)-*x. Then there is a subsequence {g„} such that gnt(z)-*x for all zeZ with at most one exception.

Proof. We can assume, without loss of generality, that z0 is the point at infinity. Since z0 e R(G), the isometric circles of all the elements of G are bounded. Let a„, a'„ be the centers of the isometric circles of gn, g~1, respectively, and let r„ be the radius of these isometric circles. Each gn then maps {z 11z — an | > r„} onto {z11z - a'tt| < r„}, and in particular, g„(z0) = a'„.By assumption, g„(z0) = a'„->x. We now pick the subsequence {g„t} so that {a„J converges to some point a. Now let z' # a be a some point of 2. Since the {gnt}are all distinct, the sequence rni converges to zero. It follows then, that for n¡ sufficiently large, | z' —ant | > r„t, and so | gnt(z') - a„'| < rni. The result now follows, since a'n -> x, and rn¡ -» 0.

II. The Combination Theorem. 6. The following theorem was proven in [5].

Theorem 1. Let Gy and G2 be Kleinian groups, and let H be a common subgroup of Gy and G2. Let Dy, D2, A be PFS'sfor Gy, G2, H, respectively. For

i = 1,2, set E¡ = Yl(D¡,H). Assume that D' = int^

n £2 n A) ± 0, and that

EyyJE2 = R(Gf) U R(G2). Then G, the group generated by Gx and G2, is Kleinian, D' is a PFSfor G, G is the free product ofGy and G2 with amalgamated

subgroup H, and, if we set D = £1n£2nA, 8*1-

then g(D) nD = 0, for all geG,


502

BERNARD MASKIT

[December

The principal aim of this paper is to find suitable conditions under which D is a FS when Dy, D2 and A are. The precise statement is as follows: Theorem 2. Let Gy and G2 be Kleinian groups. Let H be a common subgroup ofGy and G2 where H is either cyclic or consists only of the identity. Let Dy, D2,

A be FS's for Gy, G2, H, respectively. For i = 1, 2, set E¡ = Y[(D¡, H). Assume that EyUE2 = R(H), and that mt(E1nE2r\A)¥:0. Assume further that there is a simple closed curve y, contained in int(£x U £2) U L(H); y is invariant under H; the closure o/yOA is contained in int(£1n£2)> and y separates Ey —E2 and E2 —Ey. Then G, the group generated by Gy and G2 is Kleinian, G is the free product of Gy and G2 with amalgamated subgroup H, and

D = EyC\E2r\Aisa

FSfor G.

We already know, from Theorem l,that Gis Kleinian, that G is the free product of Gy and G2 with amalgamated subgroup H, and that D satisfies properties

(a) and (c) in the definition of FS; i.e. D^0,

and g(D) n D = 0 for all g e G,

g # 1. In the remainder of this chapter, we will prove that D satisfies properties

(b) and (d) in the definitionof FS; i.e. D c R(G), and ]J(D, G) = R(G). This will then complete the proof of Theorem 2. The proof of property (b) is mainly technical and simply involves chasing points around with appropriate applications of Lemma 2. After having proven that D satisfies property (b), we will then know that [~[(D,G) c R(G). We also will know that points not in R(Gy), and points not in R(G2), cannot be in R(G), and therefore that the images of these points under G also cannot be in R(G). We will then show that every other point zofL is uniquely determined by a nested sequence of images of y under G, and that

zeL(G). 7. We first observe that £t and £2 are invariant under H, and, in fact, they are precisely invariant under H; that is, if gyeGy —H, then gy(Ey)r\Ey= 0, and similarly, if g2eG2 —H, then g2(E2)C\E2 = 0. The proof of this fact is quite simple. If, for example, xegy(Ey)C\Ey, then there are points zt, z2eDx, and there are elements hx, h2eH, so that x = gyO hy(zy) = h2(z2). It follows at once that gy = h2o hy1 eH. We also observe that £x n £2 = FT(D, H). For if zeEyC\E2, then z e R(Gy) n R(G2) cz R(H), and so there is an h e H, with h(z) e A. Then, since Ey n £2 is invariant under H, h(z)eEy n £2 n A = D, and sozef](D,H).

Conversely, if z e f| (D, H), then z = h(y), where yeD = EyC\E2C\A. Again, Ey 0£2 is invariant under H, and so zeEyC\E2. Finally, we remark that since H is cyclic, L(H) consists of either 0,1, or 2 points. These simple but important remarks will be used throughout this chapter without further mention.

8. In this section, we show that D n bd(£J c R(G).


1965]

ON KLEIN'S COMBINATIONTHEOREM

Lemma 4. Let z0eEy nbd(Ey).

503

Let geGy, and let U be a connected neigh-

borhood ofz0, where U cz R(Gy). If g(U) e\zE2, then there is a point x e g(U) n y.

Proof. Since U c R(Gy), g(U) c RiGy), and so g(t/) c J?(H). If g(U) d: £2, then there are points of g(C7) which are not in £2, and, since £t U £2 = RiH), there are points of g(U) which are in Ey —£2. If g $ H, then g(z0) i Ey, and so g(z0) e E2 - Ey. If geH, then g(z0) e bd^), and so g(l/) contains points not in £x ; i.e. g(t7) contains points of £2 —Ey. Therefore g(C/) is a connected open set containing points of Ey —E2 and points of £2 —Ey. Since y separates these two sets, y passes through g(U).

Lemma 5. Let Zoeßnbd^.), U of z0, with giU) ezzE2.

and let geGy, then there is a neighborhood

Proof. Since z0eD, z0eRiGy), hence we can find a nested sequence {U„} of connected neighborhoods of z0, where each U„ ezzRiGy). Now by Lemma 4, if g({7„)d: £2, for every n, there is an x„egiU„)ny. Then lim„_00xn =g(z0)ey. This is a contradiction since y.c int(£t n E2) U L(/T), and giz0) $ L(H) U int(£t). Lemma 6. Let z0eDnbd(£,).

Then there is a neighborhood

U of z0, with

giU) y e closure(y O A) x, and then by Lemma 1, /„(z) -> x for every zey. If x' were a different point of 5, then we would have equally well that L.(z)->x', for every zey, which is an obvious contradiction. Lemma 13. IfLiH)

consists of two points, then S contains at most two points.

Proof. The proof involves picking appropriate subsequences so that various things converge. To avoid cumbersome notation, we will call the subsequence by the same name as the original sequence. Let z, z' be the two points of L(H). We first pick a subsequence so that j„(z) -> x, y'„(z')->x'. Now assume that there is some point x" e 5, x" # x,x'. Since x" e S, there is a sequence x„ e y„, with x„ -> x". Since x" # x, x', we can pick a subsequence so that, for every n, x„ ^j„iz), J„(z'). Then y„ =j~1ix„)eRiH), and so there is an n„ e H, with wn = h„iy„) e A. Observe that wn in fact lies in D n y, for y is invariant under H. We again choose a subsequence so that w„ -> w. One of the hypotheses of Theorem 2 is that closure (D Oy) c int(£t n£2), and so weRiG). We now have wn-*we K(G), and y'„o h~ 1(wn)—>x". Hence, by Lemma 2, j„o n„_1(w)->x". Now, by Lemma 3, we can choose a subsequence so that Jn ° h~Xit)-*x"> for all tell, with at most one exception. In particular, either j„o /i~1(z)->x", or j„o h~1iz')-*x". However, z and z' are fixed points of


506

BERNARD MASKIT

[December

elements of H, and so we have that either j„(z)->x", or jn(z')-*x". We have reached a contradiction since jn(z)->x, j„(z')^x', and we have assumed that

Lemma 14. IfLLH) consists of one point, then S contains at most one point.

As in the proof of Lemma 13, the proof here consists in appropriately choosing subsequences. We again use the same notation for both the subsequence and the original sequence. Let z be the one point of L(H). We first choose a subsequence so that j„(z) -* x. We now assume that there is a point x' ^ x in S. Then there is a sequence xneyn where x„-»x'. Let y„ = jn~1(x„). Since x' # x, we can choose a subsequence so that yn¥=z for all n. Since y„#z, yneR(H) and so there is an h„eH, with w„ = h„(y„) e A. Since yn ey, wne y, and so wne D n y. We choose another subsequence so that wn-* w. Since wneD C\y, weR(G). We are now in a position to apply Lemma 2, for w„-+weR(G), and jn o fo„_1(wn)= x„ -> x'. Hence /'„o h~ 1(w) -> x'. Since z is the fixed point of all elements of H, we also have that JnoK1(z)^x. Now let h ^ 1 be some fixed element of H. We set j„' =j„oh~1, K=j'n°no(j'„)~\

j'n(w) = un, j'noh(w)

= u'n, j'n(z) = vn. Then

«„ is a parabolic

Möbius transformation in G with fixed point v„, and /iA(w„)= u'„. We also know that v„-►x, m„-►x' ^ x, and, by Lemma 1, u'„-*x'. We can assume, without loss of generality that none of the points x, v„, u„, u'n is the point at infinity. Then each h„ can be uniquely represented by a matrix, with determinant + 1, of the form

f 1 + PrPn

- Pn»l '

/;„ Vn

1 - PnV„ -

Since h„(u„) = u'n, we have "n = [(1 + PAK

- P„v„] [P„"n + 1 - Pnvn] ~1-

Solving for p„, we obtain

P„ = ("„ -O

("n - vn)~HK- kT l ■

Since un, u'„-*x', v„-*x # x', we see at once that p„->-0. Then, since the v„ axe all bounded, the sequence of transformations {h„} converges to the identity. It follows then that G is not discontinuous, and we have reached a contradiction. Hence S contains at most one point. Lemmas 12, 13, and 14 exhaust all possible cases and show that S contains at most two points. Since Sis the boundary of the intersection of a nested sequence of closed discs, it follows at once that S = S = {z0} where z0 is the point of T


1965]


507

with which we started. Since 7„(z)->z0 for every zey, we have shown that z0eLiG), which is the desired conclusion. The proof of Theorem 2 is now

complete. 11. It should be remarked that, in the case that H consists only of the identity, the fact that we are dealing with Möbius transformations enters into the proof of Theorem 2 only through Lemmas 1 and 2. These lemmas remain equally valid for groups of Möbius transformations on the n-sphere S". (One regards S" as Euclidean n-space with the one point compactification, and the Möbius transformations are generated by reflections in hyperplanes, rotations, translations, and inversions in hyperspheres.) One easily sees that, for this case, the proof of Theorem 2 remains valid if one sets I = S", and y is an embedded S"-1 which separates S" into two n-discs. As a particular application of the above remark, we mention that, again for the case that H consists only of the identity, we need not restrict ourselves to orientation preserving transformations of the Riemann sphere, and we can allow transformations of the form z^iaz+

b)icz + d)~l.

III. An example. 12. In this chapter we give an example which shows that the hypotheses of Theorem 2 cannot be substantially weakened. In this example Gt and G2 are both cyclic, and H consists only of the identity. Let Gy be the group generated by the transformation at once that Dy = {z\lz%\z\ 3z. One sees

is a FS for Gy. For G2 we take the group generated by the transformation B: z -> ( —2z + 5)(z —2)_1. One easily sees that

D2= {z11z - 21= 1, | z + 21> 1,z # 1}U {z= - 3} is a FS for G2. Gy and G2, with FS's Dy and D2, satisfy some of the hypotheses of Theorem 2; Dy\JD2 = E, and intiDy nD2) ^ 0, but there is no Jordan curve lying in the interior of Dy n D2 which separates Dy —D2 and D2 —Dy. We wish to show that D = Dy nD2 is not a FS for G, the group generated by Gy and G2. Since the hypotheses of Theorem 1 are satisfiied, we know that G is discontinuous, and in fact we can prove that D to show that

RnRiG)¥=0. In order to show that R n Ä(G) ^ 0, we define new FS's D\ and D2 for Gy and G2 respectively. D[ and D2 are shown in Figures 1 and 2. These sets are


508

[December

BERNARD MASKIT

Figure 1

D'z

*■ X

Figure 2

bounded by arcs of circles orthogonal to the real axis. The inner boundary of D'y is bounded by |z\ = 1, the circles passing through 3/5 and 7/5, and the circle passing through —3/5 and —7/5. The outer boundary is obtained from the inner boundary by applying the transformation A. The boundary of D2 in the right half


1965]

509


plane is bounded by \z — 2| = 1, the circle passing through 1/2 and 3/2, and the circle passing through 8/3 and 4. The other boundary of D'2 is obtained by applying B to this boundary. One easily sees that int(D2) is symmetric with respect to the imaginary axis. One also easily sees that Gy and G2 with FS's D[ and D'2 satisfy the hypotheses of Theorem 2, and that, for example, the open interval

(4, 21/5)lies in int (Din Dereferences 1. W. Fenchel and J. Nielsen, Discontinuous groups of non-Euclidean motions, unpublished manuscript.

2. L. R. Ford, Automorphic functions, 1st ed., McGraw-Hill, New York, 1929. 3. R. Fricke and F. Klein, Vorlesungen über die Theorie der Automorphen Functionen. I,

Teubner, Leipzig, 1897. 4. F. Klein, Neue Beiträge zur Riemann'sehen Functionentheorie,

Math. Ann. 21 (1883),

141-218. 5. B. Maskit, Construction of Kleinian groups, Proc. Conf. on Complex Analysis, Minnesota,

1964. Institute for Advanced Study, Princeton, New Jersey
